Explanation

(d)
Given, $u = x^{2} + y^{2},x = s + 3t,\ y = 2s - t$
Now ,$\frac{dx}{ds} = 1,\frac{dy}{ds} = 2\ldots(i)$
$\frac{d^{2}x}{ds^{2}} = 0,\frac{d^{2}y}{ds^{2}} = 0\ldots(ii)$
Now, $u = x^{2} + y^{2}$
$\frac{du}{ds} = 2x\frac{dx}{ds} + 2y\frac{dy}{ds}$
$\frac{d^{2}u}{ds^{2}} = 2\left( \frac{dx}{ds} \right)^{2} + 2x\frac{d^{2}x}{ds^{2}} + 2\left( \frac{dy}{ds} \right)^{2} + 2y\left( \frac{d^{2}y}{ds^{2}} \right)$
$\Rightarrow \frac{d^{2}u}{ds^{2}} = 2(1)^{2} + 2x(0) + 2(2)^{2} + 2y(0) = 2 + 8 = 10$